Integral Equations and Operator Theory

, Volume 80, Issue 3, pp 379–413 | Cite as

Hörmander Functional Calculus for Poisson Estimates

Article

Abstract

The aim of the article is to show a Hörmander spectral multiplier theorem for an operator A whose kernel of the semigroup exp(−zA) satisfies certain Poisson estimates for complex times z. Here exp(−zA) acts on \({L^p(\Omega),\,1 < p < \infty}\), where \({\Omega}\) is a space of homogeneous type with the additional conditions that the volume of balls grows polynomially of exponent d and the measure of annuli is controlled by the corresponding euclidean term. In most of the known Hörmander type theorems in the literature, Gaussian bounds and self-adjointness for the semigroup are needed, whereas here the new feature is that the assumptions are the to some extent weaker Poisson bounds, and \({H^\infty}\) calculus in place of self-adjointness. The order of derivation in our Hörmander multiplier result is typically \({\frac{d}{2}}\), d being the dimension of the space \({\Omega}\). Moreover the functional calculus resulting from our Hörmander theorem is shown to be \({R}\)-bounded. Finally, the result is applied to some examples.

Keywords

Functional calculus Hörmander type spectral multiplier theorems spaces of homogeneous type Poisson semigroup 

Mathematics Subject Classification

42A45 47A60 47D03 

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© Springer Basel 2014

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques (CNRS UMR 6620)Université Blaise-Pascal (Clermont-Ferrand 2)Aubière CedexFrance

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